FUNCTIONS

1. Jun 83 7. Jan 91
x 1 2x  5 a. Given f : x  3 x  7 and
Given that f ( x)  , g (x)  ,
2x  5 x 1 4x
g:x  9 , calculate fg (10)
5x 1 5
h( x ) 
2x  1
i. Evaluate f(3) and g(4) 3x 1
b. Given that h : x  for x  ¡ ,
ii. Show that hf(3) = 3 x5
iii. Write down the expression for f—1 i. State the value of x for which
a. h( x )  0
2. Jun 87 b. h( x ) is unidentified.
x 1 ii. determine h 1 ( x)
The function g ( x ) 
x 1 Hence, solve the equation
1 3x 1
i. Calculate g ( ) 2
2 x2
Given that g ( x)  g [ g ( x )] and
2
ii. 8. Jan 92
g 3 ( x )  g[ g 2 ( x)] and so on, show that The functions f, g and h are defined as follows
g ( x )  x and hence deduce the value
2 f :x x 3 

1 g:x x 2

of g 21 ( )
2 h : x  x2  6x  9
3. Jun 88 a. Given that f 2 ( x )  f [ f ( x )] and
2x  3
Given that f : x  , g : x  2 f 3 ( x)  f [ f 2 ( x )] and so on,
x 1
i. Evaluate f (2) and gf (2)
i. Deduce an algebraic expression
in terms of x, for f 4 ( x ) and
determine f  x 
1
ii.


iii. Calculate the value of x, for which hence, calculate f 4 (3) .

a. f (x)  0 ii. Show that gf ( x )  h( x )
b. f ( x ) is unidentified.
b i. Determine the range values of
4. Jan 89 . h( x ) such that h( x )  1
Given that f : x  3x  2 , g : x  2 x  5 , ii. Show that if h( x )  25 , then
2x  3 2 x 8
h: x 


x 1
a. Evaluate c. Hence or otherwise, for x ¡ ,
i. g (6) determine the range of values of x for
ii. fg (3) which 1  h( x )  25 and represent your
b. If x) = 8, calculate the value of x.
f ( answer on a number line.
c. Obtain an expression, in terms of x, for 9. Jun 91
1
h ( x) The functions f and g are defined by:
5. Jun 89 f : x 5 x
Given that f ( x)  5 x and g ( x)  x  2 , g : x  x3
i. calculate f (2) and gf (2) Determine expressions for the functions:
ii. determine x when fg ( x )  0 i. fg ( x )
iii. prove that ( gf ) 1 23  5 ii. g 1 ( x )


6. Jan 90 10. Jan 93

x 2  14 Determine the inverse of the functions
Given that f : x  and x  0 i. f : x  2 x  5
5x
i. calculate f ( 4) x 4
g :x 


ii.
3x
ii. obtain an expression for fg ( x) if
g : x  x 1
 17. Jun 96
11. Jan 93 1
The functions f and g are defines by: If f ( x )  2 x 1 and g ( x ) 


( x  2) , calculate
2
f : x  x3
i. f (3)
g : x  px  q
ii. g 1 ( x )
a. Determine the value of x if f(x) = –64
b. Given g (0)  5 , and fg (2)  8 , iii. gf (3)
calculate the values of p and q.
c. Given that 8  fg ( x)  27 , determine 18. Jan 97
the domain of x: x
Given that f ( x )  x 2  3, and g ( x )  , find
i. If x is a real number 2
ii. If x is an integer i. the values of f (3), g (2), and fg (2)
ii. expressions for fg ( x) and gf ( x )
12. Jun 93 19. Jun 97
a composite function K is defines as f and g are functions defined as follows:
k ( x )  (2 x  1) 2 f : x  3x 5 

i. Express k ( x) as gf ( x ) , where f(x) and 1

g:x  x
g(x) are two simpler functions 2
ii. Show that k 1( x )  f 1g 1( x ) a. Calculate the value of f (3)
b. Write expressions for
i. f 1 ( x )


13. Jan 95
The functions f and g are defined as follows ii. g 1 ( x )
f ( x )  2 x 2  5, x  ¡ c. Hence, or otherwise, write an
g ( x )  3 x 2, x  ¡


expression for ( gf ) 1 ( x)


a. Evaluate 20. Jan 98

i. f(–3) Given f ( x )  3 x 2 

ii. gf(–3) i. Determine f 1( x)

b. Write an expression for g 1 ( x ) ii. Hence, solve the equation 3x  2  4
c. Determine the value of x for which 21. Jun 98
g 1
( x)  4 Given f ( x )  x 2 and g ( x)  5 x  3 , calculate


d. Write an expression for gf ( x ) i. f (2)

ii. gf (2)
14. Jun 95 iii. g 1 ( x )


1
Given that f ( x )  x, and g ( x )  x - 2 calculate
2
22. Jan 99
i. g (2)
Given that f ( x )  2 x 3 

ii. fg (4)
1 (4)
i. Determine an expression for f 1( x )
iii. f


ii. Hence, or otherwise, calculate the value

of x for which f ( x )  7
15. Resit 95
Given that f ( x )  4  5 x and g ( x )  x 2  1, 23. Jun 99
calculate If h( x )  1  3 x and k ( x )  x  2 , calculate
i. f (2)
i. hk ( x)
ii. gf ( 1) 

ii. hk (4)
1
iii. f (4) iii. (hk )1 ( x)
iv. the value of x when hk ( x )  0
16. Jan 96 24. Jan 00
x 2  16 1
Given that h ( x )  , calculate The function f : x  x 1
x2 2
i. h( 2) i. Find the value of f  0
ii. the value of x for which h( x)  0
ii. Find the value of x for which f  x   5 
25. Jun 00 30. Jan 03
Given that
Given that g  x   6  x and h  x   x3
f : x 3x
x2 i. h  3

g:x 
x 5 ii. hg  2 
a. Calculate g  2  iii. gh  2 
b. State the value of x for which g  x  is
not defined 31. Jun 03
c. Derive an expression for gf  x  Two functions g and h are defined as
d. Calculate the value of f 1  4 2x  3 1
g:x and h : x 
x 4 

x
26. Jan 01 Calculate

Given that f  x   x  2 and g  x  

3
,
i. the value of g  7 
x ii. the value for which g  x   6
i. calculate f  1 Write expressions for
ii. write an expression for gf  x  iii. hg  x 
iii. calculate the values ox x so that iv. g 1  x 
f  x  g  x 
27. Jun 01 32. Jan 04
Given that g  x   x  3 and h  x   x , calculate
2
Given that f  x   3 x  4 and g  x   x ,
i. g  5 calculate
i. g  25 
ii. g 1  7 
ii. gf  15
iii. hg  0 
33. Jan 05
28. Jan 02 Te functions f and g are such that
Given that f  x   9  x, and g  x   x 2 , 2x  5
f  x  and g  x   2 x 3 .


calculate x 4
i. f  3  Calculate the value of
ii. g  4  i. g  4 

fg  2  ii. fg  2 
iii.
iii. g 1  7 
29. Jun 02
The functions f and g are defined by 34. Jun 05
x The functions f and g are defined by
f  x  1 1
3 f  x  x  5 g  x   x2
g  x  2 x 1 2
Evaluate
a. Calculate g  3  i. g  3   g  3 

b. Find in its simplest form 1

ii. f  6
i. f 1
 x


iii. fg  2 
ii.  x
g 1


iii. fg  x 
iv.  fg  1  x 


c. Show that  fg  1  x   g 1 f 1  x 
